Seconded on the moving out part. It IS good your parents are actually trying to help you. It may be in their own way, which isn't really helping you after all, but at least they are trying.
And getting caught in your type of situation is always to be expected, no matter how well anyone believes it may be hidden. The best course of action is to head off their catching on and just tell them beforehand and in a controlled manner. Unfortunately, even with your last near catch you didn't do that. So at this point all you can really do is focus on damage control.
You can actually use your situation to your advantage though, and slowly turn your current lying to your parents into the truth. Start off by following their advice and go see a doctor. With patient confidentiality you can tell the doc you are happy managing periodic bedwetting with diapers, but want to rule out any condition that could be harmful to you. Maybe even decline any pills, but see if you can get a prescription for better diapers than just depends.
Go back to your parents and show them the prescription. Use the depends they got you for now though (double diaper them at least). Also, go get a mattress protector for your new mattress I'm sure you will be getting. Tell your parents it's for just in case, because you don't want to risk "ruining" another mattress (at least in their eyes). Also iterate your current wetting has only been periodic, maybe linked to stress more than any amount you do or don't drink.
After 2-3 days, tell your parents the depends keep leaking really bad. Say you've already looked into getting better diapers, and have ordered some to try out (which as long as they aren't with prints you should order a bag of the same kind you're using).
In the end, your parents will be almost forced by the doctors note/prescription into accepting diapers are what you need. And the kind you need are what you're already having delivered to their house. They will be under the impression you need them, which is true to a degree. Only they won't realize your need is a mental one, not physical. This will also leave it open for you to not use every diaper, and possibly even open to not wearing them when you don't feel like it.